Let's say we are looking at the saturated water table and we want to figure out the
saturation pressure at a temperature of 53 degrees C. There is data at 50 degrees and
saturation pressure data at 55 degrees C but nothing at 53. One thing that we can do is
linearly interpolate between those two values. What I mean by linearly interpolate is here
we are at 50 degrees C and there is some saturation pressure and at 55 degrees C there is some
other saturation pressure. In this case the pressure is a little bit higher. Let's draw
a line between the two. We are going to assume a linear dependence between those two. Here
are the two data points that we have on the table. The saturation pressures at 50 and
55 degrees C. We drew a line between it and we want to figure out what is the saturation
pressure when T is equal to 53 degrees C. 53 degrees C would be somewhere right here
between the two. We want to figure out what is P_sat at this location. The equation I
drew up here is a general form for linear interpolation between two known data points.
The y in this case is analogous to our saturation pressure and x is analogous to the saturation temperature.
Let's call this point 1 and we will call this point up here point 2. Here is the analogous
expression with pressures and temperatures. What I will do is plug in P1 and P2 and T1
and T2 and plug in a value for T and we will come up with the saturation pressure that
we are interested in at a temperature of 53 degrees C. When I plug in numbers I get a
pressure of 14.4 kPa. To check that I did it right it is between 12.352 and 15.8 kPa.
This assumes that there is a linear dependence between point 1 and point 2. I will show you
that that is not rigorously correct but that is the best that we can do at the moment.
To demonstrate this let's recreate these numbers in EES and I am going to evaluate at 50 and
55 degrees. Here are two commands in EES that will recreate the tables in the back of your
book. Instead of using water it uses a function called steam IAPWS. That is a more accurate
equation of state than using either water or using steam for example. Let's use steam
IAPWS to recreate the steam tables themselves. I will run it and what I come up with is P1_sat
of 12.352 kPa and P2_sat of 15.763. Those are the exact same numbers for P_sat for point
1 and P_sat for point 2. You would think, well, alright, I'll evaluate at 50 and 55 and I'll evaluate this now at 53 and
what I would expect would be 14.4 kPa. Here I have a third line in EES and finding the pressure. The same steam IAPWS.
temperature 53 degrees C. And by the way, I'm using a quality of 50 percent arbitrarily in EES.
I could just as well do a quality of 0 or quality of 1 if I wanted to. I am just telling
EES that I am working with a saturated liquid. Here is the P_sat. What I come up with is
14.3. This is the actual value. This is the value if you wanted to be rigorously correct
this is what I would choose. It turns out that instead of being a linear relationship
between 1 and 2 there is a little bit of curvature here. I will exaggerate it in this case. But
the real data look like this and the actual value if we don't use linear interpolation the exact
value is about 14.3 kPa. So linear interpolation makes our life easy but there are small errors
associated with that.